Regression calibration using the BLUP technique functions almost equivalently in the case of our generalized error models as in a standard error model. We model \(\widehat{X}_i = \widehat{E}(X_i\mid X_i^*, Z_i)\)  as a linear function and then estimate \(\widehat{\Theta }_\text{RC}\)  by solving \(0 = U_n(Y,\widehat{X},Z; \widehat{\Theta }_\text{RC})\) . When assuming an additive measurement error model with replicate observations, we take the mean of the replicates as \(X_i^*\) . In the case of non-identically distributed measurements, it is unlikely that this will be the most efficient combination. Intuitively, measurements with lower variance ought to contribute more to our proxy measure than those with high variance. We define \(X_i^* = \sum _{j=1}^{\kappa _i} \alpha _jX_{ij}^*\) , for some set of weights \(\lbrace \alpha _j\rbrace \)  such that \(\sum _{j=1}^{\kappa _i} \alpha _j = 1\) , with \(\alpha _j \ge 0\)  for \(j=1,\hdots ,\kappa _i\) . Taking \(\kappa _i = k\) , \(\alpha _j = 1/k\) , and \(J_{01} = \lbrace 1,\hdots ,k\rbrace \) , simplifies to the standard estimators [1]}.
